Maximum, minimum, optimum

The notion of optimum often evokes that of maximum or minimum of a function which, like a fluid, would evolve in a continuous manner, and even differentiable. The arsenal of analysis and differential calculus then comes to mind. But optimization also often concerns quantities that can take only a finite or countable number of values. If the range of techniques from analysis is then of no use, discrete mathematics and graph theory take over. It is the computer, a sequential machine best suited to combinatorial environments, that will then most often be called upon to solve optimum problems.

Optimization also applies to geometric contexts. Nature was the first to seek it, whether in the shape of honeycomb cells or in that of certain landforms shaped by water or wind. Mathematicians, too, have devoted themselves to it, now using the tools of differential calculus, which have supplanted the ancient methods to the point of making geometry unrecognizable. The very questions themselves have been transformed by the increased efficiency offered by these methods. But not everything reduces to differential calculus. When combinatorial aspects intrude, problems can prove arduous, to the point that certain seemingly simple statements, like that of the Heilbronn triangles, still resist the sagacity of researchers.

While it is often possible to theoretically demonstrate the existence of an optimal solution to a given problem, an explicit and complete form of this solution is more difficult to obtain outside of a few textbook cases. It is therefore necessary to settle for calculating an estimate of it using algorithms. Examples are numerous, both in a "psychological" context (optimization of collective well-being using the Pareto model) and in physics. Thus, when the quantity to be optimized varies like a fluid, clever procedures can reach the precious minimum or maximum, provided care is taken not to remain stuck in a local pocket: it is the global optimum that is sought!